DOCSLIB.ORG

Brief Derivation of the Heston Model

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Brief Derivation of the Heston Model Simpli…ed Derivation of the Heston Model by Fabrice Douglas Rouah www.FRouah.com www.Volopta.com Note: A complete treatment of the Heston model, including a more detailed derivation of what appears below, is available in the forthcoming book "The Heston Model and its Extensions in Matlab and C#", available September 3, 2013 from John Wiley & Sons. The table of contents is available at www.FRouah.com. The stochastic volatility model of Heston [2] is one of the most popular equity option pricing models. This is due in part to the fact that the Heston model produces call prices that are in closed form, up to an integral that must evaluated numerically. In this Note we present a complete derivation of the Heston model. 1 Heston Dynamics The Heston model assumes that the underlying, St; follows a Black-Scholes type stochastic process, but with a stochastic variance vt that follows a Cox, Ingersoll, Ross process. Hence dSt = rStdt + pvtStdW1;t (1) dvt = ( vt)dt + pvtdW2;t E [dW1;tdW2;t] = dt: We will drop the time index and write S = St; v = vt;W1 = W1;t; and W2 = W1;t for notional convenience. 2 The Heston PDE In this section we explain how to derive the PDE from the Heston model. This derivation is a special case of a PDE for general stochastic volatility models which is described by Gatheral [1]. Form a portfolio consisting of one option V = V (S; v; t), units of the stock S, and units of another option U = U(S; v; t) that is used to hedge the volatility. The portfolio has value = V + S + U where = t. Assuming the portfolio is self-…nancing, the change in portfolio value is d = dV + dS + dU: 1 2.1 Portfolio Dynamics Apply It¯o’sLemma to dV . We must di¤erentiate with respect to the variables t; S; and v. Hence @V @V @V 1 @2V 1 @2V @2V dV = dt + dS + dv + vS2 dt + 2v dt + vS dt: @t @S @v 2 @S2 2 @v2 @v@S Applying It¯o’sLemma to dU produces the identical result, but in U. Combining these two expressions, we can write the change in portfolio value, d; as d = dV + dS + dU (2) @V 1 @2V @2V 1 @2V = + vS2 + vS + 2v dt + @t 2 @S2 @v@S 2 @v2 @U 1 @2U @2U 1 @2U + vS2 + vS + 2v dt + @t 2 @S2 @v@S 2 @v2 @V @U @V @U + + dS + + dv: @S @S @v @v 2.2 The Riskless Portfolio In order for the portfolio to be hedged against movements in the stock and against volatility, the last two terms in Equation (2) involving dS and dv must be zero. This implies that the hedge parameters must be @V @v = @U (3) @v @U @V = : @S @S Moreover, the portfolio must earn the risk free rate, r. Hence d = rdt. Now with the values of and from Equation (3) the change in value of the riskless portfolio is @V 1 @2V @2V 1 @2V d = + vS2 + vS + 2v dt + @t 2 @S2 @v@S 2 @v2 @U 1 @2U @2U 1 @2U + vS2 + vS + 2v dt @t 2 @S2 @v@S 2 @v2 which we write as d = (A + B) dt: Hence we have A + B = r (V + S + U) : Substituting for and re-arranging, produces the equality @V @U A rV + rS @S B rU + rS @S @V = @U (4) @v @v which we exploit in the next section. 2 2.3 The PDE in Terms of the Price The left-hand side of Equation (4) is a function of V only, and the right-hand side is a function of U only. This implies that both sides can be written as a function f(S; v; t) of S; v, and t. Following Heston, specify this function as f(S; v; t) = ( v) + (S; v; t); where (S; v; t) is the price of volatility risk. Substitute f(S; v; t) into the left-hand side of Equation (4), substitute for B, and rearrange to produce the Heston PDE expressed in terms of the price S @U 1 @2U @2U 1 @2U + vS2 + vS + 2v (5) @t 2 @S2 @v@S 2 @v2 @U @U rU + rS + [( v) (S; v; t)] = 0: @S @v This is Equation (6) of Heston [2]. 2.4 The PDE in Terms of the Log Price Let x = ln S and express the PDE in terms of x; t and v instead of S; t; and v. This leads to a simpler form of the PDE. We need the following derivatives, which are straightforward to derive @U @2U @2U ; ; @S @v@S @S2 Plug into the Heston PDE Equation (5). All the S terms cancel and we obtain the Heston PDE in terms of the log price x = ln S @U 1 @2U 1 @U @2U + v + r v + v + (6) @t 2 @x2 2 @x @v@x 1 @2U @U 2v rU + [( v) v] = 0 2 @v2 @v where, as in Heston, we have written the market price of risk to be a linear function of the volatility, so that (S; v; t) = v. 3 The Call Price The call price is of the form r + CT (K) = e E (ST K) (7) xt r = e P1(x;h v; ) e i KP2(x; v; ): In this expression Pj(x; v; ) each represent the probability of the call expiring in-the-money, conditional on the value xt = ln St of the stock and on the value vt of the volatility at time t, where = T t is the time to expiration. Now take the following derivatives of C using (7). These are straightforward to obtain @C @C @2C @C @2C @2C ; ; ; ; ; : @t @x @x2 @v @v2 @x@v 3 We use these derivatives in the following section. 3.1 The PDE for P1 and P2 The call price C in Equation (7) follows the PDE in Equation (6), which we write here in terms of C but using the time derivative with respect to rather than t @C 1 @2C 1 @C @2C + v + r v + v + (8) @ 2 @x2 2 @x @v@x 1 @2C @C 2v rC + [( v) v] = 0: 2 @v2 @v The derivatives of C from (7) will be in terms of P1 and P2. Substitute these derivatives into the PDE (8) and regroup terms common to P1. Set K = 0 and S = 1 to obtain the PDE for P1. Now regroup terms common to P2 and set S = 0;K = 1; and r = 0 to obtain the PDE for P2 For notional convenience, combine the PDEs for P1 and P2 into a single expression @P @2P 1 @2P 1 @2P j + v j + v j + v2 j (9) @ @x@v 2 @x2 2 @v2 @Pj @Pj + (r + ujv) + (a bjv) = 0 @x @v 1 1 for j = 1; 2 and where u1 = 2 ; u2 = 2 ; a = ; b1 = + ; and b2 = +. This is Equation (12) of Heston [2] but in terms of rather than t. That explains the minus sign in the …rst term of Equation (9) above. 3.2 Obtaining the Characteristic Functions Heston assumes that the characteristic functions for the logarithm of the termi- nal stock price, x = ln ST , are of the form fj(; x; v) = exp (Cj (; ) + Dj (; ) v0 + ix) (10) where Cj and Dj are coe¢ cients and = T t is the time to maturity. The characteristic functions fj will follow the PDE in Equation (9). This is a con- sequence of the Feynman-Kac theorem. Hence the PDE for the characteristic function is, from Equation (9) @f @2f 1 @2f 1 @2f j + v j + v j + v2 j (11) @ @x@v 2 @x2 2 @v2 @fj @fj + (r + ujv) + (a bjv) = 0: @x @v To evaluate this PDE for the characteristic function we need the following deriv- atives, which are straightforward to derive @f @f @2f @f @2f @2f j ; j ; j ; j ; j ; j : @ @x @x2 @v @v2 @v@x 4 Substitute these derivatives in Equation (11), drop the fj terms, and re-arrange to obtain two di¤erential equations @Dj 1 2 1 2 2 = iDj + D + uji bjDj (12) @ 2 2 j @C j = ri + aD : @ j These are Equations (A7) in Heston [2]. Heston speci…es the initial conditions Dj(0; ) = 0 and Cj(0; ) = 0. The …rst Equation in (12) is a Riccati equation in Dj while the second is an ODE for Cj that can solved using straightforward integration once Dj is obtained. 3.3 Solving the Heston Riccati Equation From Equation (12) the Heston Riccati equation is @Dj 2 = Pj QjDj + RD (13) @ j The corresponding second order ODE is w00 + Qjw0 + PjR = 0 (14) The solution to the Heston Riccati equation (13) is therefore 1 K e + e Dj = (15) R Ke + e Using the initial condition Dj(0; ) = 0 produces the solution for Dj dj bj i + dj 1 e Dj = 2 d : (16) 1 gje j where 2 2 2 dj = (i bj) 2uji : q bj i + dj gj = : bj i dj The solution for Cj is found by integrating the second equation in (12). Hence dj y Qj + dj 1 e Cj = ridy + a d y dy + K1 (17) 2R 1 gje j Z0 Z0 where K1 is a constant. Integrate and apply the initial condition Cj (0; ) = 0, and substitute for dj;Qj; and gj to obtain the solution for Cj dj a 1 gje Cj = ri + 2 (bj i + dj) 2 ln : (18) 1 gj where a = .
Load more

Recommended publications
  • Parameter Estimation in Stochastic Volatility Models Via Approximate Bayesian Computing
    Parameter Estimation in Stochastic Volatility Models Via Approximate Bayesian Computing A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Achal Awasthi, B.S. Graduate Program in Department of Statistics The Ohio State University 2018 Master's Examination Committee: Radu Herbei,Ph.D., Advisor Laura S. Kubatko, Ph.D. c Copyright by Achal Awasthi 2018 Abstract In this thesis, we propose a generalized Heston model as a tool to estimate volatil- ity. We have used Approximate Bayesian Computing to estimate the parameters of the generalized Heston model. This model was used to examine the daily closing prices of the Shanghai Stock Exchange and the NIKKEI 225 indices. We found that this model was a good fit for shorter time periods around financial crisis. For longer time periods, this model failed to capture the volatility in detail. ii This is dedicated to my grandmothers, Radhika and Prabha, who have had a significant impact in my life. iii Acknowledgments I would like to thank my thesis supervisor, Dr. Radu Herbei, for his help and his availability all along the development of this project. I am also grateful to Dr. Laura Kubatko for accepting to be part of the defense committee. My gratitude goes to my parents, without their support and education I would not have had the chance to study worldwide. I would also like to express my gratitude towards my uncles, Kuldeep and Tapan, and Mr. Richard Rose for helping me transition smoothly to life in a different country.
    [Show full text]
  • Multi-Period Public Transport Design: a Novel Model and Solution Approaches
    S. Gelareh, S. Nickel Multi-period public transport design: A novel model and solution approaches Berichte des Fraunhofer ITWM, Nr. 139 (2008) © Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM 2008 ISSN 1434-9973 Bericht 139 (2008) Alle Rechte vorbehalten. Ohne ausdrückliche schriftliche Genehmigung des Herausgebers ist es nicht gestattet, das Buch oder Teile daraus in irgendeiner Form durch Fotokopie, Mikrofilm oder andere Verfahren zu reproduzieren oder in eine für Maschinen, insbesondere Datenverarbei tungsanlagen, ver- wendbare Sprache zu übertragen. Dasselbe gilt für das Recht der öffentlichen Wiedergabe. Warennamen werden ohne Gewährleistung der freien Verwendbarkeit benutzt. Die Veröffentlichungen in der Berichtsreihe des Fraunhofer ITWM können bezogen werden über: Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM Fraunhofer-Platz 1 67663 Kaiserslautern Germany Telefon: +49 (0) 6 31/3 16 00-0 Telefax: +49 (0) 6 31/3 16 00-10 99 E-Mail: [email protected] Internet: www.itwm.fraunhofer.de Vorwort Das Tätigkeitsfeld des Fraunhofer-Instituts für Techno- und Wirtschaftsmathematik ITWM umfasst anwendungsnahe Grundlagenforschung, angewandte Forschung sowie Beratung und kundenspezifische Lösungen auf allen Gebieten, die für Tech- no- und Wirtschaftsmathematik bedeutsam sind. In der Reihe »Berichte des Fraunhofer ITWM« soll die Arbeit des Instituts konti- nuierlich einer interessierten Öffentlichkeit in Industrie, Wirtschaft und Wissen- schaft vorgestellt werden. Durch die enge Verzahnung mit dem Fachbereich Ma- thematik der Universität Kaiserslautern sowie durch zahlreiche Kooperationen mit internationalen Institutionen und Hochschulen in den Bereichen Ausbildung und Forschung ist ein großes Potenzial für Forschungsberichte vorhanden. In die Be- richtreihe sollen sowohl hervorragende Diplom- und Projektarbeiten und Disser- tationen als auch Forschungsberichte der Institutsmitarbeiter und Institutsgäste zu aktuellen Fragen der Techno- und Wirtschaftsmathematik aufgenommen werden.
    [Show full text]
  • Forecasting Security's Volatility Using Low-Frequency Historical Data
    Forecasting security’s volatility using low-frequency historical data, high-frequency historical data and option-implied volatility Huiling Yuana, Yong Zhoub,c, Zhiyuan Zhangd, Xiangyu Cuid,∗ aSchool of Statistics and Information, Shanghai University of International Business and Economics. bInstitute of Statistics and Interdisciplinary Sciences and School of Statistics, Faculty of Economics and Management, East China Normal University. cAcademy of Mathematics and Systems Sciences, Chinese Academy of Sciences. dSchool of Statistics and Management, Shanghai University of Finance and Economics. Abstract Low-frequency historical data, high-frequency historical data and option data are three major sources, which can be used to forecast the underlying security’s volatility. In this paper, we propose two econometric models, which integrate three information sources. In GARCH-Itˆo-OI model, we assume that the option-implied volatility can influence the security’s future volatility, and the option-implied volatility is treated as an observable exogenous variable. In GARCH- Itˆo-IV model, we assume that the option-implied volatility can not influence the security’s volatility directly, and the relationship between the option-implied volatility and the security’s volatility is constructed to extract useful information of the underlying security. After providing the quasi-maximum likelihood estimators for the parameters and establishing their asymptotic properties, we also conduct a series of simulation analysis and empirical analysis to compare the arXiv:1907.02666v1 [q-fin.ST] 5 Jul 2019 proposed models with other popular models in the literature. We find that when the sampling interval of the high-frequency data is 5 minutes, the GARCH-Itˆo-OI model and GARCH-Itˆo-IV model has better forecasting performance than other models.
    [Show full text]
  • The Heston Model - Stochastic Volatility and Approximation - ‌
    B.Sc. Thesis The Heston Model - Stochastic Volatility and Approximation - ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ Author Patrik Karlsson, [email protected] Supervisor Birger Nilsson (Department of Economics, Lund University) Abstract The crude assumption on log normal stock returns and constant volatility in the Black-Scholes model is a big constraint which constructs smile and skew inconsistent prices. The Heston model and its suggested approximation built on stochastic volatility are introduced and faced against the Black-Scholes model in hope of producing option prices where the smile and skew are taken into account.. As one will observe later on is that numerical calculation and approximation of the Heston model will provide us with more accurate calculations. ‌ ‌ Keywords Black-Scholes, Derivative Pricing, Heston, Monte Carlo, Volatility Smile. List of notation cdf - Normal cumulative density function. E [X] - Expected value of s.v. X. f (·) - Normal pdf. F (·) - Normal cdf. X Ft - Filtration, all information about X until time t. GBM - Geometric Brownian motion. K - Strike price. L - Likelihood function. MC - Monte Carlo. N (0; 1) - Normal distribution with mean 0 and variance 1. P - Historical measure. PDE - Partial differential equation. pdf - Probability density function. φ (·) - Normal pdf. Φ - Payoff. Π(S; t) - Derivative value, with underlying asset S at time t. Q - Risk neutral martingale measure. Rd - (d × 1) −dimensional real value. rt - Interest rate at time t. σ - Volatility. SDE - Stochastic differential equation. s.v. - Stochastic variable 3 St - Stock Value at time t. T - Time to maturity. θ- Parameter set. Θ - Parameter space. W - Wiener process, standard N (0; 1). ⊆ - Subset. - End of proof or derivation.
    [Show full text]
  • 126 FM12 Abstracts
    126 FM12 Abstracts IC1 which happens in applications to barrier option pricing or Optimal Execution in a General One-Sided Limit structural credit risk models. In this talk, I will present Order Book novel adaptive discretization schemes for the simulation of stopped Lvy processes, which are several orders of magni- We construct an optimal execution strategy for the pur- tude faster than the traditional approaches based on uni- chase of a large number of shares of a financial asset over form discretization, and provide an explicit control of the a fixed interval of time. Purchases of the asset have a non- bias. The schemes are based on sharp asymptotic estimates linear impact on price, and this is moderated over time by for the exit probability and work by recursively adding dis- resilience in the limit-order book that determines the price. cretization dates in the parts of the trajectory which are The limit-order book is permitted to have arbitrary shape. close to the boundary, until a specified error tolerance is The form of the optimal execution strategy is to make an met. initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is Peter Tankov set to match the order book resiliency. At the end of this Universit´e Paris-Diderot (Paris 7) period, another lump purchase is made, and following that [email protected] there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase.
    [Show full text]
  • Valuing a European Option with the Heston Model Yuan Yang
    Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 5-10-2013 Valuing a European option with the Heston model Yuan Yang Follow this and additional works at: http://scholarworks.rit.edu/theses Recommended Citation Yang, Yuan, "Valuing a European option with the Heston model" (2013). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Valuing a European Option with the Heston Model Rochester Institute of Technology School of Mathematical Sciences College of Science Applied & Computational Mathematics Program Master’s Thesis Applicant‟s Name: Yuan Yang Proposed Defense Date: 05/10/2013 Advisor‟s Name: Dr. Bernard Brooks Committee Member 1: Dr. Raluca Felea Committee Member 2: Dr. James Marengo Graduate Program Director: Dr. Tamas Wiandt - 1 - Valuing a European Option with the Heston Model Valuing a European Option with the Heston Model A thesis present by Yuan Yang to The School of the Mathematical Sciences in partial fulfillment of the requirements for the degree of Master of Science in the subject of Applied and Computational Mathematics Rochester Institute of Technology Rochester, New York May 2013 - 2 - Valuing a European Option with the Heston Model Abstract In spite of the Black-Scholes (BS) equation being widely used to price options, this method is based on a hypothesis that the volatility of the underlying is a constant.
    [Show full text]
  • GARCH Option Pricing Models and the Variance Risk Premium
    Journal of Risk and Financial Management Article GARCH Option Pricing Models and the Variance Risk Premium Wenjun Zhang 1,* and Jin E. Zhang 2 1 Department of Mathematical Sciences, School of Engineering, Computer and Mathematical Sciences, Auckland University of Technology, Auckland 1142, New Zealand 2 Department of Accountancy and Finance, Otago Business School, University of Otago, Dunedin 9054, New Zealand; [email protected] * Correspondence: [email protected] Received: 3 February 2020; Accepted: 4 March 2020; Published: 9 March 2020 Abstract: In this paper, we modify Duan’s (1995) local risk-neutral valuation relationship (mLRNVR) for the GARCH option-pricing models. In our mLRNVR, the conditional variances under two measures are designed to be different and the variance process is more persistent in the risk-neutral measure than in the physical one, so that one is able to capture the variance risk premium. Empirical estimation exercises show that the GARCH option-pricing models under our mLRNVR are able to price the SPX one-month variance swap rate, i.e., the CBOE Volatility Index (VIX) accurately. Our research suggests that one should use our mLRNVR when pricing options with GARCH models. Keywords: GARCH option-pricing models; stochastic volatility; the CBOE VIX; variance risk premium JEL Classification: G13; C52 1. Introduction In this paper, we modify the local risk-neutral valuation relationship (mLRNVR) in the GARCH option-pricing models. The GARCH option-pricing model was first introduced by Duan(1995) with a locally risk-neutral valuation relationship (LRNVR), in which the conditional variances and model parameters remained the same under the physical measure and the risk-neutral measure.
    [Show full text]
  • The Mean-Reverting 4/2 Stochastic Volatility Model: Properties and Financial Applications
    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 2-3-2021 10:00 AM The Mean-Reverting 4/2 Stochastic Volatility Model: Properties And Financial Applications Zhenxian Gong, The University of Western Ontario Supervisor: Escobar-Anel, Marcos, The University of Western Ontario A thesis submitted in partial fulfillment of the equirr ements for the Doctor of Philosophy degree in Statistics and Actuarial Sciences © Zhenxian Gong 2021 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Applied Statistics Commons, Other Applied Mathematics Commons, Probability Commons, and the Statistical Models Commons Recommended Citation Gong, Zhenxian, "The Mean-Reverting 4/2 Stochastic Volatility Model: Properties And Financial Applications" (2021). Electronic Thesis and Dissertation Repository. 7686. https://ir.lib.uwo.ca/etd/7686 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract In this thesis, we target commodity and volatility index markets, and develop a novel stochastic volatility model that incorporates mean-reverting property and 4/2 stochastic volatility process. Commodities and volatility indexes have been shown to be mean-reverting. The 4/2 stochastic volatility process integrates two processes that have contrary behaviors. As a result, not only is the 4/2 stochastic volatility process able to reproduce “smile” and “skew”, but also model the asset price time series very well even in the most extreme situation like financial crisis where the assets’ prices become highly volatile.
    [Show full text]
  • Regime-Switching Stochastic Volatility Model: Estimation and Calibration
    Regime-switching Stochastic Volatility Model : Estimation and Calibration to VIX options Stéphane Goutte, Amine Ismail, Huyên Pham To cite this version: Stéphane Goutte, Amine Ismail, Huyên Pham. Regime-switching Stochastic Volatility Model : Esti- mation and Calibration to VIX options. 2017. hal-01212018v2 HAL Id: hal-01212018 https://hal.archives-ouvertes.fr/hal-01212018v2 Preprint submitted on 1 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Regime-switching Stochastic Volatility Model : Estimation and Calibration to VIX options Stéphane GOUTTE1,2,∗, Amine ISMAIL3,5 and Huyên PHAM3,4 1Université Paris 8, LED , 2 rue de la Liberté, 93526 Saint-Denis Cedex, France. 2 PSB, Paris School of Business, 59 rue Nationale 75013 Paris, France. 3 Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, Universités Paris 7 Diderot. 4CREST-ENSAE 5Natixis, 47 Quai d’Austerlitz, 75013 Paris. Abstract We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynam- ics with the instantaneous variance driven by a Cox-Ingersoll-Ross (CIR) process and each parameter modulated by a hidden Markov chain.
    [Show full text]
  • Indian Institute of Management Calcutta Working Paper Series
    Indian Institute of Management Calcutta Working Paper Series WPS No. 796 March 2017 Comparison of Black Scholes and Heston Models for Pricing Index Options Binay Bhushan Chakrabarti Retd. Professor, Indian Institute of Management Calcutta D. H. Road, Joka, P.O. Kolkata 700 104 https://www.iimcal.ac.in/faculty/publications/working-papers/archive/2017 Arijit Santra PGP Student, Indian Institute of Management Calcutta D. H. Road, Joka, P.O. Kolkata 700 104 COMPARISON OF BLACK SCHOLES AND HESTON MODELS FOR PRICING INDEX OPTIONS ArijitSantra Indian Institute of Management Calcutta Email id: [email protected] Faculty Advisor: Retired. Prof. B.B. Chakarabarti, Finance & Control ABSTRACT:This paper studies the performance of Heston Model and Black-Scholes Model in pricing index options. I have compared the two models based on 1074 call option prices of S&P 500 on1st November, 2016. I have calibrated the parameters of the Heston Model by non-linear least square optimization using call option prices from a period of 20 days (3rd October, 2016 to 31st October, 2016). The in-sample data had a total of 25,392 call options and thus 20 strike prices for each time- to-maturity. We observe that both Heston Model and Black Scholes Model under-price in-the-money options and over-price out-of-the money options, but the degree of error is different.Black Scholes Model slightly outperforms Heston Model for short term ITM, DITM and ATM options where Heston Model is unable to capture the high implied volatility.But Heston Model starts to give better estimates for ITM, DITM and ATM options as the time-to-maturity increases.
    [Show full text]
  • Asymptotics of the Time-Discretized Log-Normal SABR Model: the Implied Volatility Surface
    Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface Dan Pirjol 1, Lingjiong Zhu 2 March 11, 2020 Abstract We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model. 1 Introduction The method of the asymptotic expansions has been used in the literature to study the properties of stochastic volatility models under a wide variety of strike-maturity regimes. The short maturity limit at fixed strike for the implied volatility was first derived for the SABR model in the celebrated work of [23]. This model is defined by the two-dimensional stochastic differential equation: dS(t) = σ(t)Sγ(t)dW (t) ; (1.1) dσ(t) = !σ(t)dZ(t) ; where W (t) and Z(t) are two standard Brownian motions, with E[dW (t)dZ(t)] = %dt.
    [Show full text]
  • Gamma Expansion of the Heston Stochastic Volatility Model
    Noname manuscript No. (will be inserted by the editor) Gamma Expansion of the Heston Stochastic Volatility Model Paul Glasserman ¢ Kyoung-Kuk Kim Received: date / Accepted: date Abstract We derive an explicit representation of the transitions of the Heston stochastic volatil- ity model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of in¯nite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the Pitman-Yor decomposition of Bessel bridges. We combine this representation with the Broadie-Kaya exact simulation method and use it to circumvent the most time-consuming step in that method. Keywords Stochastic Volatility model ¢ Monte Carlo methods Mathematics Subject Classi¯cation (2000) 60H35 ¢ 65C05 ¢ 91B70 JEL Classi¯cation C63 ¢ G12 ¢ G13 1 Introduction The Heston [18] stochastic volatility model is among the most fundamental models in both the theory and practice of ¯nancial mathematics. It provides a natural extension beyond geometric Brownian motion as a description of asset price dynamics, modeling volatility as a positive, mean- reverting, stochastic process. It also o®ers some tractability: Heston [18] showed that standard European options can be priced in this model through an extension of the Black-Scholes formula; evaluation of this formula reduces to numerical inversion of a characteristic function. Despite this degree of tractability, the Heston model has proved notoriously di±cult to sim- ulate accurately.
    [Show full text]